Linearized Poisson-Boltzmann Equation Research Articles

Energetics of the H-Bond Network in Exiguobacterium sibiricum Rhodopsin.

Exiguobacterium sibiricum rhodopsin (ESR) functions as a light-driven proton pump utilizing Lys96 for proton uptake and maintaining its activity over a wide pH range. Using a combination of methodologies including the linear Poisson-Boltzmann equation and a quantum mechanical/molecular mechanical approach with a polarizable continuum model, we explore the microscopic mechanisms underlying its pumping activity. Lys96, the primary proton uptake site, remains deprotonated owing to the loss of solvation in the ESR protein environment. Asp85, serving as a proton acceptor group for Lys96, does not form a low-barrier H-bond with His57. Instead, deprotonated Asp85 forms a salt-bridge with protonated His57, and the proton is predominantly located at the His57 moiety. Glu214, the only acidic residue at the end of the H-bond network exhibits a pKa value of ∼6, slightly elevated due to solvation loss. It seems likely that the H-bond network [Asp85···His57···H2O···Glu214] serves as a proton-conducting pathway toward the protein bulk surface.

Exact solution of electroosmotic flow of the second grade fluid in a circular microchannel driving by an AC electric field

Microfluidics development has led to increased attention on electroosmotic flow in biotechnology and lab-on-a-chip technologies. In this particular investigation, an analytical solution is derived for the electroosmotic flow of a second grade fluid in a circular microchannel driven by an AC electric field. The derivation is based on the Debye–Hückel linear approximation. Analytical expressions for the electric potential and velocity profile are obtained by solving the linearized Poisson–Boltzmann equation and the Navier–Stokes equation. Numerical results are presented to examine the influence of various factors on the velocity profile and volumetric flow rate. Specifically, the effects of the retardation time λ, the frequency of the applied external electric field ω, and the electrokinetic width K are analyzed. The results demonstrate that a more considerable retardation time λ leads to reduced fluctuations in the velocity profile and a smaller maximum velocity value. Conversely, smaller values of λ or ω result in larger amplitudes of the volumetric flow rate.

Coupling finite and boundary element methods to solve the Poisson-Boltzmann equation for electrostatics in molecular solvation.

The Poisson-Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the linearised Poisson-Boltzmann equation, where the finite element method is applied in a confined solute region and the boundary element method in the external solvent region. As a proof-of-concept exercise, we use the simplest methods available: Johnson-Nédélec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. As validation, we compare against well-established finite difference solvers for an extensive binding energy data set, and with the finite difference code APBS (to 0.5%) for Gaussian permittivities. We also show scaling results from protein G B1 (955 atoms) up to immunoglobulin G (20,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large-sized problems on a single workstation. The development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments that consider space-varying field parameters, and mixed linear-nonlinear schemes for molecular electrostatics with implicit solvent models.

A Mechano-Electro-Chemical Coupling Model for Bending Analysis of Single-Stranded DNA Microbeam Biosensors Due to Flexoelectricity

Abstract Highly compliant structures such as microbeams can deform substantially in response to interactions between molecules adsorbed on their surface. To understand such systems and improve their detection signals, a mechano-electro-chemical coupling model for mechanical deformations of the microbeams immobilized single-stranded DNA (ssDNA) is established due to flexoelectricity. The governing equations and corresponding boundary conditions of ssDNA microbeams are derived by using the variational principle. The bending deformations of ssDNA microbeams (one for cantilever beam and another for simply supported beam) are derived. The electric potential in the regions inside and outside the ssDNA layer is obtained by linear Poisson–Boltzmann equation for different electrolyte solutions. The analytical expressions to quantify the beam deflection and the potential difference of ssDNA layer are presented. The theoretical predictions are compared with the experimental data to validate the applicability of the present model. Numerical results reveal that the solution types, thickness, and elastic modulus of substrate materials have an obvious influence on the deflections of ssDNA microbeams. Therefore, the present model can help to improve the reading of the bending deformation signal of the microbeam biosensors.

The electrokinetic flow of a micropolar fluid in a microtube with velocity and spin velocity slippage

This article examines a theoretical investigation of electrokinetic microstructure fluid flow in a circular microtube. It is based on a linearized Poisson–Boltzmann equation and a microstructure electrolyte of Eringen micropolar type. In our analysis, we assumed that the surface zeta potential was small due to the use of the Debye-Hückel approximation. The influences of velocity slip and spin velocity slip at the surface of the microtube, as well as overlapped electrical double layers are considered. The findings of the present study demonstrate that the micropolarity of fluid particles has a declining effect on volume transport, streaming potential, and energy transport efficiency, while velocity slip and spin velocity slip enhance these factors. It was found that when the ionic Péclet number decreases from 1 to 0.1, the volume transport decreases approximately 56%, the streaming potential decreases by around 334.6%, and the energy transport efficiency increases by approximately 88.2%. The physical quantities of this study are highlighted through graphs for different values of the relevant parameters and compared with the available data in the literature.

Topological impact of nanopore electrodes on the structure of the electrical double layer and the differential capacitance

The electrical double layer for three different topologies of nanopore electrodes is studied, i.e., the interior and exterior electrical double layers of planar, cylindrical and spherical nanopores immersed into a point-ions electrolyte, and not connected to a power source, are analytically attained through the linearized Poisson-Boltzmann equation. Thus, analytical formulas for the mean electrostatic potential, electrolyte's reduced concentration, and electrical field profiles, are exhibited. Their corresponding analytical expressions for their differential capacitances are presented. All the nanopores are treated as permeable, so the electrolyte outside and inside the electrodes are at the same chemical potential. Analogous analytical formulas for solid nano-electrodes are obtained as a corollary of those for nanopores. In particular, their analytical expressions for the differential capacitance here derived are shown to be consistent with the capacitive compactness proposed in the past by one of us. Numerical results of all of the above functions are analyzed as a function of the nanopores geometrical parameters and the electrolyte's temperature and molar concentration. It is found that the spherical topology, at lower temperatures, has the higher differential capacitance. It is demonstrated that for the three nanopore topologies here considered their capacitances reduce to that of a single planar electrode, in the limit of infinitely wide nanopores. The electrical double layer and mean electrostatic potential of the three topologies are in qualitatively agreement with those from the non-linearized Poisson-Boltzmann, hypernetted chain/mean-spherical approximation (HNC/MSA) equations and computer simulations results presented in the past, within the low mean electrostatic potential assumption.

A kernel-free boundary integral method for the nonlinear Poisson-Boltzmann equation

This work proposes a boundary integral formulation based Cartesian grid method for the nonlinear Poisson-Boltzmann (PB) interface problem in biophysics. The method (a) does not have the limitation associated with the standard finite difference method for complex interfaces, (b) avoids the generation of any unstructured volume or surface grids needed by the finite element method, and (c) solves the nonlinear and variable coefficient PDE in the framework of boundary integral equations. The method solves the nonlinear PB equation with the Newton iterative method. It first reformulates the linearized PB equation in each Newton iteration as a Fredholm system of two boundary integral (BI) equations of the second kind, which is well-conditioned, and then solves the BI system with a Krylov subspace method. The evaluation of boundary and volume integrals involved in the solution of the BI system is done with a kernel-free method, which does not need to know Green's function associated with the variable coefficient PB equation. The kernel-free method evaluates volume and boundary integrals by solving equivalent simple interface problems on Cartesian grids with either a fast Fourier transform based Poisson solver or a geometric multigrid preconditioned conjugate gradient solver, either of which involves computational work essentially linearly proportional to the number of nodes on the Cartesian grid. Numerical examples in both two and three space dimensions are presented to demonstrate the efficiency and accuracy of the proposed numerical method.

Grand canonical DFT based constant charge method for electrochemical deuterium/hydrogen evolution reaction micro-kinetics on Pt (1 1 1)

Applied external potential is considered to have significant impact on the reaction mechanisms and kinetic behaviors for electrochemical reactions. The electrochemical isotope deuterium/hydrogen (D/H) evolution shows significant different kinetic behavior under applied potential conditions, but the intrinsic mechanisms from the electrochemical isotope effect remains unclear. In this work, the grand canonical density functional theory (GC-DFT) based constant charge method was used to systematically investigate the electrochemical D/H evolution reactions (DER/HER) on the Pt (111) surface. By using this method, the linear and quadratic relationship between charge distribution (q), electronic energy (Eelect) and applied potential (U) were observed respectively, which are resulted from solving of linear Poisson-Boltzmann equation and Kohn-Sham-Mermin equations in combination of solvent modification. Equilibrium Pourbaix diagrams of D/H adsorbing on Pt (111) surface was subsequently investigated to revealed the electrode structure in real reaction condition, with θD/H coverages on the cathode surface calculated to be 0.0 monolayer (ML) and 1/6 ML for alkaline and acidic solvent at −1.23 V to 0.0 V vs. SHE. Mechanistic studies of HER/DER on the Pt (111) surface revealed that Volmer-Tafel reaction pathway is more favorable than the Volmer-Heyrovsky pathway. Analysis of the applied potential effect on the reaction energy barriers show that the acidic Volmer step is very sensitive toward the applied potential condition, with the structure of transition state likely to shift. Comparisons of the calculated current–potential polarization curves between HER and DER revealed that in the acidic condition, the calculated current density of HER is close to DER, revealing that the H*/D* migration and coupling on the electrode is comparable to each other when the solution is saturated with H3O+/D3O+. In the alkaline reaction condition, discrepancies of j value between DER and HER becomes slightly larger, revealing that jointly control effect from Volmer and Tafel step was enhanced. This work provides the new insight for investigating the microkinetic properties of electrochemical isotope reaction.

Interaction of Nanoparticles in Electrolyte Solutions.

The interaction between nanoparticles includes several components; however, the most frequently used are electrostatic, caused by overlapping double electrical layers, and London-van der Waals interactions, caused by quantum and thermodynamic fluctuations of electromagnetic fields. Only these two kinds of interaction are considered below. The electrostatic interaction is calculated based on the linearized Poisson-Boltzmann equation for particles with constant electrical potential of the surfaces (constant ζ potentials). An exact solution of the problem is obtained for both identical particles and particles of different sizes. For the London-van der Waals interaction, the screening of static fluctuations and the retardation of electromagnetic fields for the dispersive part of the interaction are taken into account. The total interaction energy for two particles was calculated for a range of possible nanoparticle sizes from 1 to 103 nm and electrolyte concentration from 10-2 to 10-6 mol/L. The predominance of the London-van der Waals force over the shielded electrostatic repulsion force was found at high electrolyte concentrations in the range from 10-2 to 10-3 mol/L at large interparticle distances.

Redox Potentials of Iron-Sulfur Clusters in Type I Photosynthetic Reaction Centers.

The electron transfer pathways in type I photosynthetic reaction centers, such as photosystem I (PSI) and reaction centers from green sulfur bacteria (GsbRC), are terminated by two Fe4S4 clusters, FA and FB. The protein structures are the basis of understanding how the protein electrostatic environment interacts with the Fe4S4 clusters and facilitates electron transfer. Using the protein structures, we calculated the redox potential (Em) values for FA and FB in PSI and GsbRC, solving the linear Poisson-Boltzmann equation. The FA-to-FB electron transfer is energetically downhill in the cyanobacterial PSI structure, while it is isoenergetic in the plant PSI structure. The discrepancy arises from differences in the electrostatic influences of conserved residues, including PsaC-Lys51 and PsaC-Arg52, located near FA. The FA-to-FB electron transfer is slightly downhill in the GsbRC structure. Em(FA) and Em(FB) exhibit similar levels upon isolation of the membrane-extrinsic PsaC and PscB subunits from the PSI and GsbRC reaction centers, respectively. The binding of the membrane-extrinsic subunit at the heterodimeric/homodimeric reaction center plays a key role in tuning Em(FA) and Em(FB).

Entropy generation on EMHD transport of couple stress fluid with slip-dependent zeta potential under electrokinetic effects

This study theoretically explores the electroosmotic and electromagnetohydrodynamic (EMHD) transport of couple stress fluid in a microchannel subjected to an applied magnetic and electric field simultaneously to predict the flow dynamics, thermal transport, and entropy generation in a thermofluidic system with slip-dependent (SD) zeta potential. Solving the linearized Poisson-Boltzmann equation gives a closed-form solution for the slip-dependent (SD) electrical potential distribution established in the Electrical Double Layer (EDL). Using suitable boundary conditions and the SD zeta potential, analytical expressions for flow velocity, temperature, and entropy generation rate are determined. There are a variety of important parameters that have been considered, including the couple stress, slip length, Hartmann number, pressure gradient, and transverse electric field. The findings for each of these parameters are graphically displayed. After accounting for the SD zeta potential for the slip parameter and Hartmann number, the fluid velocity is shown to rise; however, the normalized velocity for the couple stress parameter exhibits the reverse trend owing to its strong resistive influence. It is noticed that with a higher Brinkman number, the magnitude value of the temperature is larger for SD zeta potential than for slip-independent (SI) zeta potential owing to an increase in viscous dissipation effect. The results reveal that the flow velocity and thermal energy distribution have a substantial effect on the quantity of entropy generated in the channel under SD zeta potential.

Dynamical aspects of transient electro-osmotic flow of Burgers' fluid with zeta potential in cylindrical tube

Abstract In this article, we consider the flow of a Burgers’ fluid of transient electro-osmotic type in a small tube with a circular cross-section. Analytical results are found for the transient velocity and, electric potential profile by solving the Navier–Stokes and the linearized Poisson–Boltzmann equations. Moreover, these equations are solved with the help of the integral transform method. We consider cases in which the velocity of the fluid changes with time and those in which the velocity of the fluid does not change with time. Furthermore, special results for classical fluids such as Newtonian, second grade, Maxwell, and Oldroyd-B fluids are obtained as the particular cases of the outcomes of this work and that these results actually strengthen the results of the solution. This study of the nonlinear problem of Burgers’ fluid in a specified physical model will help us to understand the behavior of blood clotting and the block of any kind of problem in which this type of fluid is used. The solution of the complex velocity profile of Burgers’ fluid will help us in the future to solve the problems in which this transient electro-osmotic type of small tube is involved. At the end, numerical results are shown graphically that can help us to understand the complex behavior of the Burgers’ fluid, and also the analysis of the Burgers’ fluid shows dissimilarity with other fluids that are considered in this work.

Energetic Diversity in the Electron-Transfer Pathways of Type I Photosynthetic Reaction Centers.

Photosynthetic reaction centers from heliobacteria (HbRC) and green sulfur bacteria (GsbRC) are homodimeric proteins and share a common ancestor with photosystem I (PSI), classified as type I reaction centers. Using the HbRC crystal structure, we calculated the redox potential (Em) values in the electron-transfer branches, solving the linear Poisson-Boltzmann equation and considering the protonation states of all titratable sites in the entire protein-pigment complex. Em(A-1) for bacteriochlorophyll g at the secondary site in HbRC (-1157 mV) is as low as Em(A-1) for chlorophyll a in PSI (-1173 mV). Em(A0/HbRC) is at the same level as Em(A0/GsbRC) and is 200 mV higher than Em(A0/PSI) due to the replacement of PsaA-Trp697/PsaB-Trp677 in PSI with PshA-Arg554 in HbRC. In contrast, Em(FX) for the Fe4S4 cluster in HbRC (-420 mV) is significantly higher than Em(FX) in GsbRC (-719 mV) and PSI (-705 mV) due to the absence of acidic residues that correspond to PscA-Asp634 in GsbRC and PsaB-Asp575 in PSI. It seems likely that type I reaction centers have evolved, adopting (bacterio)chlorophylls suitable for their light environments while maintaining electron-transfer cascades.

A Cartesian FMM-accelerated Galerkin boundary integral Poisson-Boltzmann solver

The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method for solving the linear Poisson-Boltzmann (PB) equation. The solver has combined advantages in accuracy, efficiency, and memory usage as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(N) Cartesian Fast Multipole Method (FMM) to accelerate the GMRES iteration. In addition, special numerical treatments such as adaptive FMM order, block diagonal preconditioners, Galerkin discretization, and Duffy's transformation are combined to improve the performance of the solver, which is validated on benchmark Kirkwood's sphere and a series of testing proteins.

Electro-osmotic flow of generalized Maxwell fluids in triangular microchannels based on distributed order time fractional constitutive model

Based on the linearized Poisson–Boltzmann equation, the electro-osmotic flow of a generalized Maxwell fluid under an alternating field in an isosceles right triangle microchannel is studied. The finite volume method and L2 interpolation method are used to obtain the numerical solution. An analytical solution is constructed to verify the accuracy of the numerical solution. Under the alternating current, the velocity will oscillate periodically. The velocity amplitude of the Maxwell fluid with the distributed order time fractional derivative is larger than that of Newtonian fluids and fractional Maxwell fluids, which indicates that its elastic characteristics further promote fluid flow. However, oscillation of the velocity does not achieve synchronization with the oscillation of the electric fields. Furthermore, due to the existence of the angle effect, the velocity will develop at acute angles and form a larger value of velocity first. The numerical results show that the relaxation time, electrokinetic width, zeta potential, and angular Reynolds number play important roles in determining the velocity and amplitude of electro-osmosis.

Arbitrary-Shape Dielectric Particles Interacting in the Linearized Poisson-Boltzmann Framework: An Analytical Treatment.

This work considers the interaction of two dielectric particles of arbitrary shape immersed into a solvent containing a dissociated salt and assuming that the linearized Poisson-Boltzmann equation holds. We establish a new general spherical re-expansion result which relies neither on the conventional condition that particle radii are small with respect to the characteristic separating distance between particles nor on any symmetry assumption. This is instrumental in calculating suitable expansion coefficients for the electrostatic potential inside and outside the objects and in constructing small-parameter asymptotic expansions for the potential, the total electrostatic energy, and forces in ascending order of Debye screening. This generalizes a recent result for the case of dielectric spheres to particles of arbitrary shape and builds for the first time a rigorous (exact at the Debye-Hückel level) analytical theory of electrostatic interactions of such particles at arbitrary distances. Numerical tests confirm that the proposed theory may also become especially useful in developing a new class of grid-free, fast, highly scalable solvers.

The electrostatic potential inside and around α-lactalbumin: Fluctuations and mean-field models.

The electrostatic potential (EP) generated by the protein α-lactoalbumin in the presence of added salt is computed as a thermal average at a given point in space. With this aim, constant pH Monte Carlo (MC) simulations are performed within the primitive model, namely, the solvent is treated as a continuum dielectric. The study of the thermal and spatial fluctuations of the EP reveals that they are remarkably high inside the protein. The calculations indicate that fluctuations inside the protein are mainly due to the asymmetric distribution of the charge groups, while the charge fluctuations of the titratable groups play a minor role. The computed EP matches very well with the one obtained from the Poisson equation for the average charge density in spherical symmetry. The Tanford-Kirkwood multipole expansion reproduces the simulated angular-averaged potential rather accurately. Surprisingly, two of the simplest mean-field models, the linear Poisson-Boltzmann (PB) equation and Donnan potential, provide good estimations of the average EP in the effective protein surface (surface EP). The linear PB equation predicts a linear relationship between charge and surface EP, which is numerically reproduced only if the small ions within the protein are taken into account. On the other hand, the partition coefficients of the small ions inside and outside the protein predicted by Donnan theory reproduce reasonably well the simulation results.

Energetics of the Electron Transfer Pathways in the Homodimeric Photosynthetic Reaction Center.

Photosynthetic reaction centers from a green sulfur bacterium (GsbRC), the PscA/PscA proteins, and photosystem I (PSI), PsaA/PsaB proteins, share structural similarities. Here, we report the redox potential (Em) values of GsbRC by solving the linear Poisson-Boltzmann equation and considering the protonation states of all titratable sites in the entire GsbRC protein and identify the factors that shift the Em values with respect to PSI. The Em values for one-electron reduction of the accessory (A-1) and adjacent (A0) chlorophylls in GsbRC are 100-250 mV higher than those in PSI, whereas the Em values for the Fe4S4 cluster (FX) are at the same level. The PsaA-Trp697/PsaB-Trp677 pair in PSI, which forms the A1-quinone binding site, is replaced with PscA-Arg638 in GsbRC. PsaB-Asp575 in PSI, which is responsible for the Em difference between A1A and A1B quinones in PSI, is absent in GsbRC. These discrepancies also contribute to the upshift in Em(A-1) and Em(A0) in GsbRC with respect to PSI. It seems likely that the upshifted Em for chlorophylls in GsbRC ultimately originates from the characteristics of the electrostatic environment that corresponds to the A1 site of PSI.

Transient Analysis of the Electro-Osmotic Flow of Multilayer Immiscible Maxwell Fluids in an Annular Microchannel

This work investigates the transient multilayer electro-osmotic flow of viscoelastic fluids through an annular microchannel. The dimensionless mathematical model of multilayer flow is integrated by the linearized Poisson-Boltzmann equation, the Cauchy momentum equation, the rheological Maxwell model, initial conditions, and the electrostatic and hydrodynamic boundary conditions at liquid-liquid and solid-liquid interfaces. Although the main force that drives the movement of fluids is due to electrokinetic effects, a pressure gradient can also be added to the flow. The semi-analytical solution for the electric potential distribution and velocity profiles considers analytical techniques as the Laplace transform method, with numerical procedures using the inverse matrix method for linear algebraic equations and the concentrated matrix exponential method for the inversion of the Laplace transform. The results presented for velocity profiles and velocity tracking at the transient regime reveal an interesting oscillatory behavior that depends on elastic fluid properties via relaxation times. The time required for the flow to reach steady-state is highly dependent on the viscosity ratios and the dimensionless relaxation times. In addition, the influence of other dimensionless parameters on the flow as the electrokinetic parameters, zeta potentials at the walls, permittivity ratios, ratio of pressure forces to electro-osmotic forces, number of fluid layers, and annular thickness are investigated. The findings of this study have significant implications for the precise control of parallel fluid transport in microfluidic devices for flow-focusing applications.

Modeling of the Electrostatic Interaction and Catalytic Activity of [NiFe] Hydrogenases on a Planar Electrode.

Hydrogenases are a group of enzymes that have caught the interest of researchers in renewable energies, due to their ability to catalyze the redox reaction of hydrogen. The exploitation of hydrogenases in electrochemical devices requires their immobilization on the surface of suitable electrodes, such as graphite. The orientation of the enzyme on the electrode is important to ensure a good flux of electrons to the catalytic center, through an array of iron-sulfur clusters. Here we present a computational approach to determine the possible orientations of a [NiFe] hydrogenase (PDB 1e3d) on a planar electrode, as a function of pH, salinity, and electrode potential. The calculations are based on the solution of the linearized Poisson-Boltzmann equation, using the PyGBe software. The results reveal that electrostatic interactions do not truly immobilize the enzyme on the surface of the electrode, but there is instead a dynamic equilibrium between different orientations. Nonetheless, after averaging over all thermally accessible orientations, we find significant differences related to the solution's salinity and pH, while the effect of the electrode potential is relatively weak. We also combine models for the protein adsoption-desorption equilibria and for the electron transfer between the proteins and the electrode to arrive at a prediction of the electrode's activity as a function of the enzyme concentration.